Answer: [tex](-1+\sqrt{19}, \ \ 0), \ \ (-1-\sqrt{19}, \ \ 0)[/tex]
==============================================
Explanation:
The center is (h,k) = (-1,7)
A point on the circle is (x,y) = (-3,-1)
Plug these four items into the circle template equation below. This will help us find r^2. We don't need to solve for r itself.
[tex](x-h)^2 + (y-k)^2 = r^2\\\\(-3-(-1))^2 + (-1-7)^2 = r^2\\\\(-3+1)^2 + (-1-7)^2 = r^2\\\\(-2)^2 + (-8)^2 = r^2\\\\4 + 64 = r^2\\\\r^2 = 68[/tex]
The circle equation [tex](x-h)^2 + (y-k)^2 = r^2\\\\[/tex] updates to [tex](x+1)^2 + (y-7)^2 = 68\\\\[/tex]
------------------------
To find the x intercept(s), replace y with 0 and isolate x.
[tex](x+1)^2 + (y-7)^2 = 68\\\\(x+1)^2 + (0-7)^2 = 68\\\\(x+1)^2 + 49 = 68\\\\(x+1)^2 = 68-49\\\\(x+1)^2 = 19\\\\x+1 = \pm\sqrt{19}\\\\x = -1\pm\sqrt{19}\\\\x = -1+\sqrt{19} \ \text{ or } \ x = -1-\sqrt{19}\\\\[/tex]
One x intercept is located at [tex](-1+\sqrt{19}, \ \ 0)[/tex] while the other one is located at [tex](-1-\sqrt{19}, \ \ 0)[/tex]
All x intercepts have a y coordinate of 0, as this is a location where the curve crosses the x axis.
